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880

The smallest squares containing k 880's :
38809 = 197²,
3988048801 = 63151²,
188018806880025 = 13711995².

880² + 881² + 882² + ... + 8033² = 415443².

Page of Squares : First Upload December 26, 2005 ; Last Revised October 2, 2006
by Yoshio Mimura, Kobe, Japan

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881

The smallest squares containing k 881's :
11881 = 109²,
88190881 = 9391²,
1418816881881 = 1191141².

The squares which begin with 881 and end in 881 are
88190881 = 9391²,   8815519881 = 93891²,   88125265881 = 296859²,
88144265881 = 296891²,   881046926881 = 938641²,...

881² = 776161, a square with 3 kinds of digits.

881² = 50³ + 64³ + 73³.

881⁷ = 4 1 1 9 3 7 5 2 8 3 6 0 8 6 6 18 8 5 6 1, and
4²+1²+1²+9²+3²+7²+5²+2²+8²+3²+6²+0²+8²+6²+6²+18²+8²+5²+6²+1² = 881.

881² = 1! + 7! + 7! + 8! + 9! + 9!

(1² + 8)(3² + 8)(7² + 8)(9² + 8) = 881² + 8.

881² + 882² + 883² + ... + 6929² = 332695²,
881² + 882² + 883² + ... + 903² = 4278².

3-by-3 magic squares consisting of different squares with constant 881²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(9, 172, 864, 332, 801, 156, 816, 324, 73),	(9, 224, 852, 588, 636, 161, 656, 567, 156),
(24, 207, 856, 431, 744, 192, 768, 424, 81),	(28, 471, 744, 609, 548, 324, 636, 504, 343),
(33, 276, 836, 564, 649, 192, 676, 528, 201),	(48, 399, 784, 504, 656, 303, 721, 432, 264),
(84, 289, 828, 393, 756, 224, 784, 348, 201),	(96, 404, 777, 588, 609, 244, 649, 492, 336),
(105, 244, 840, 460, 735, 156, 744, 420, 215),	(116, 432, 759, 543, 564, 404, 684, 521, 192),
(136, 471, 732, 564, 612, 289, 663, 424, 396)

881² = 776161, 7⁴ + 7⁴ + 6⁴ + 1⁴ + 6⁴ + 1⁴ = 86²,
881² = 776161, 7 + 76 + 16 + 1 = 10²,
881² = 776161, 77 + 6 + 16 + 1 = 10²,
881² = 776161, 776 + 1 + 6 + 1 = 28².

Page of Squares : First Upload December 26, 2005 ; Last Revised October 9, 2009
by Yoshio Mimura, Kobe, Japan

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882

The smallest squares containing k 882's :
88209 = 297²,
18882882225 = 137415²,
2488298823888201 = 49882851².

882² = 3³ + 57³ + 84³.

882² = (4² + 4)(199² + 4) = (1² + 5)(11² + 5)(32² + 5) = (1² + 5)(2² + 5)(3² + 5)(32² + 5)
= (2² + 5)(3² + 5)(4² + 5)(17² + 5) = (3² + 5)(7² + 5)(32² + 5) = (4² + 5)(11² + 5)(17² + 5).

882^k + 2478^k + 16338^k + 24402^k are squares for k = 1,2,3 (210², 29484², 4348260²).

The integral triangle of sides 689, 16810, 17493 has square area 882².

Komachi equations:
882² = 98² * 7² / 6² * 54² * 3² / 21² = 98² / 7² / 6² * 54² / 3² * 21².

(1 + 2 + ... + 6)(7 + 8 + ... + 20)(21 + 22 + ... + 28) = 882²,
(1 + 2 + 3)(4 + 5 + ... + 24)(25 + 26 + ... + 38) = 882²,
(1 + 2 + 3)(4 + 5 + ... + 17)(18 + 19 + ... + 45) = 882²,
(1 + 2 + 3)(4 + 5 + ... + 10)(11 + 12 + ... + 73) = 882².

(1² + 2² + ... + 76²)(77² + 78² + ... + 440²)(441² + ... + 882²) = 29128169070².

The 4-by-4 magic squares consisting of different squares with constant 882:

882² = 777924, 7 + 7 + 7 + 9 + 2 + 4 = 6².

Page of Squares : First Upload December 26, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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883

The smallest squares containing k 883's :
8836 = 94²,
288388324 = 16982²,
388368883883929 = 19707077².

883² = 779689, a square every digit of which is greater than 5.

883² = 1³ + 10³ + 92³,

The square root of 883 is 29. 7 1 5 3 1 5 9 1 6 2 0 7 2 5 13 8 8 11 8 7 ..., and
29² = 7² + 1² + 5² + 3² + 1² + 5² + 9² + 1² + 6² + 2² + 0² + 7² + 2² + 5² + 13² + 8² + 8² + 11² + 8² + 7².

3-by-3 magic squares consisting of different squares with constant 883²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(17, 402, 786, 606, 577, 282, 642, 534, 287),	(18, 186, 863, 593, 642, 126, 654, 577, 138),
(18, 242, 849, 298, 801, 222, 831, 282, 98),	(18, 486, 737, 534, 593, 378, 703, 438, 306),
(33, 222, 854, 438, 746, 177, 766, 417, 138),	(33, 350, 810, 550, 642, 255, 690, 495, 242),
(42, 495, 730, 530, 570, 417, 705, 458, 270),	(54, 273, 838, 458, 726, 207, 753, 422, 186),
(81, 402, 782, 478, 639, 378, 738, 458, 159),	(82, 417, 774, 513, 654, 298, 714, 422, 303),
(177, 334, 798, 402, 753, 226, 766, 318, 303),	(222, 431, 738, 462, 702, 271, 719, 318, 402)

883² = 779689, 7 + 7 + 9 + 68 + 9 = 10².

Page of Squares : First Upload December 26, 2005 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

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884

The smallest squares containing k 884's :
14884 = 122²,
884884009 = 29747²,
2588488418884 = 1608878².

The squares which begin with 884 and end in 884 are
88433674884 = 297378²,   884310782884 = 940378²,   884769746884 = 940622²,
8840976730884 = 2973378²,   8842427798884 = 2973622²,...

884² = 781456, a square with different digits.

884² = 781456, 781 + 4 + 56 = 29².

Page of Squares : First Upload December 26, 2005 ; Last Revised October 2, 2006
by Yoshio Mimura, Kobe, Japan

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885

The smallest squares containing k 885's :
788544 = 888²,
50885885241 = 225579²,
2885068858588569 = 53712837².

Komachi equation: 885² = 1³ + 23³ - 4³ * 5³ + 6³ * 7³ + 89³.

1/ 885 = 0.00 1 1 2 9 9 4 3 5 0 2 8 2 4 8 5 8 7 5 7 0 6 2 1 4 6 8 9 2 6 5 ...,
the sum of the squares of its digits is 885 : 1² + 1² + 2² + 9² + ... +6² + 5² = 885.

3-by-3 magic squares consisting of different squares with constant 885²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(9, 162, 870, 450, 750, 135, 762, 441, 90),	(19, 92, 880, 208, 856, 85, 860, 205, 40),
(20, 212, 859, 485, 716, 188, 740, 475, 100),	(20, 301, 832, 560, 640, 245, 685, 532, 176),
(20, 485, 740, 560, 580, 365, 685, 460, 320),	(40, 205, 860, 460, 740, 155, 755, 440, 140),
(40, 260, 845, 299, 800, 232, 832, 275, 124),	(40, 260, 845, 520, 691, 188, 715, 488, 184),
(40, 352, 811, 460, 685, 320, 755, 436, 152),	(40, 460, 755, 565, 568, 376, 680, 499, 268),
(40, 565, 680, 596, 520, 397, 653, 440, 404),	(43, 320, 824, 376, 740, 307, 800, 365, 100),
(54, 153, 870, 222, 846, 135, 855, 210, 90),	(64, 245, 848, 595, 640, 140, 652, 560, 211),
(77, 536, 700, 580, 560, 365, 664, 427, 400),	(85, 496, 728, 560, 595, 340, 680, 428, 371),
(90, 345, 810, 615, 558, 306, 630, 594, 183),	(91, 188, 860, 340, 805, 140, 812, 316, 155),
(100, 365, 800, 475, 700, 260, 740, 400, 275),	(100, 421, 772, 475, 628, 404, 740, 460, 155),
(128, 365, 796, 400, 740, 275, 779, 320, 272),	(135, 330, 810, 450, 729, 222, 750, 378, 279),
(140, 392, 781, 440, 715, 280, 755, 344, 308),	(140, 440, 755, 595, 608, 244, 640, 469, 392),
(152, 461, 740, 565, 520, 440, 664, 548, 205),	(155, 460, 740, 604, 485, 428, 628, 580, 229),
(205, 440, 740, 484, 688, 275, 712, 341, 400)

Page of Squares : First Upload December 26, 2005 ; Last Revised July 9, 2010
by Yoshio Mimura, Kobe, Japan

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886

The smallest squares containing k 886's :
478864 = 692²,
4688688676 = 68474²,
34788668868864 = 5898192².

Komachi square sum : 886² = 5² + 73² + 269² + 841².

590² + 591² + 592² + ... + 886² = 12804².

The square root of 886 is 29. 7 6 5 7 5 2 13 22 ...,
and 29² = 7² + 6² + 5² + 7² + 5² + 2² + 13² + 22² = 29².

886² = 784996, 7 + 84 + 9 + 96 = 14²,
886² = 784996, 7 + 84 + 99 + 6 = 14².

Page of Squares : First Upload December 26, 2005 ; Last Revised October 2, 2006
by Yoshio Mimura, Kobe, Japan

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887

The smallest squares containing k 887's :
887364 = 942²,
29488788729 = 171723²,
22887887988879225 = 151287435².

887² = 786769, a zigzag square.

887² = 786769, a square every digit of which is greater than 5.

887² = 14⁴ + 22⁴ + 22⁴ + 23⁴.

887 is the first integer which can be written as the sum of a prime and a square in 11 ways:
2² + 883, 8² + 823, 10² + 787, 12² + 743, 14² + 691, 16² + 631, 18² + 563, 20² + 487, 24² + 311, 26² + 211, 28² + 103.

3-by-3 magic squares consisting of different squares with constant 887²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(3, 94, 882, 486, 738, 77, 742, 483, 54),	(3, 302, 834, 426, 762, 157, 778, 339, 258),
(3, 302, 834, 454, 717, 258, 762, 426, 157),	(3, 426, 778, 454, 717, 258, 762, 302, 339),
(3, 426, 778, 498, 643, 354, 734, 438, 237),	(14, 93, 882, 522, 714, 67, 717, 518, 66),
(14, 333, 822, 387, 742, 294, 798, 354, 157),	(22, 186, 867, 354, 797, 162, 813, 342, 94),
(22, 354, 813, 582, 643, 186, 669, 498, 302),	(22, 438, 771, 582, 589, 318, 669, 498, 302),
(42, 221, 858, 338, 798, 189, 819, 318, 122),	(42, 534, 707, 626, 483, 402, 627, 518, 354),
(45, 438, 770, 490, 630, 387, 738, 445, 210),	(67, 258, 846, 582, 626, 237, 666, 573, 122),
(78, 346, 813, 606, 573, 302, 643, 582, 186),	(90, 237, 850, 445, 750, 162, 762, 410, 195),
(99, 482, 738, 582, 522, 419, 662, 531, 258),	(130, 330, 813, 435, 738, 230, 762, 365, 270),
(141, 302, 822, 402, 762, 211, 778, 339, 258)

887² = 786769, 7 + 86 + 7 + 69 = 13²,
887² = 786769, 78 + 6 + 76 + 9 = 13².

Page of Squares : First Upload December 26, 2005 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

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888

The smallest squares containing k 888's :
88804 = 298²,
97888888384 = 312872²,
988858588838884 = 31446122².

888² + 889² + 890² + ... + 57826² = 8028399².

138^k + 417^k + 582^k + 888^k are squares for k = 1,2,3 (45², 1149², 31185²).

888² = 788544, 7 + 8 + 8 + 5 + 4 + 4 = 6²,
888² = 788544, 7 + 8 + 8 + 54 + 4 = 9²,
888² = 788544, 7 + 8 + 85 + 44 = 12²,
888² = 788544, 7 + 88 + 5 + 44 = 12²,
888² = 788544, 78 + 8 + 54 + 4 = 12²,
888² = 788544, 7 + 885 + 4 + 4 = 30².

888² = 788544 appears in the decimal expression of π:
  π = 3.14159•••788544••• (from the 31609th digit).

Page of Squares : First Upload December 26, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

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889

The smallest squares containing k 889's :
6889 = 83²,
5537889889 = 74417²,
9488968893889 = 3080417².

The squares which begin with 889 and end in 889 are
8898714889 = 94333²,   88903559889 = 298167²,   889092468889 = 942917²,
889405544889 = 943083²,   889563989889 = 943167²,...

889² = 790321, a square with different digits.

889² = 3⁰ + 3¹ + 3⁴ + 3⁶ + 3⁷ + 3⁹ + 3¹⁰ + 3¹¹ + 3¹².

A cubic polynomial :
(X + 209²)(X + 528²)(X + 684²) = X³ + 889²X² + 403788²X + 75480768².

889² + 890² + 891² + ... + 9097² = 500749².

3-by-3 magic squares consisting of different squares with constant 889²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(12, 151, 876, 209, 852, 144, 864, 204, 47),	(12, 279, 844, 591, 628, 216, 664, 564, 177),
(16, 132, 879, 564, 681, 92, 687, 556, 96),	(36, 87, 884, 128, 876, 81, 879, 124, 48),
(36, 425, 780, 600, 564, 335, 655, 540, 264),	(39, 264, 848, 432, 736, 249, 776, 423, 96),
(60, 236, 855, 495, 720, 164, 736, 465, 180),	(81, 492, 736, 624, 556, 303, 628, 489, 396),
(88, 369, 804, 561, 648, 236, 684, 484, 297),	(96, 367, 804, 529, 624, 348, 708, 516, 151),
(111, 304, 828, 576, 657, 164, 668, 516, 279),	(124, 456, 753, 489, 668, 324 732, 369, 344),
(126, 371, 798, 462, 714, 259, 749, 378, 294),	(144, 344, 807, 488, 711, 216, 729, 408, 304),
(177, 396, 776, 524, 681, 228, 696, 412, 369),	(204, 556, 663, 591, 408, 524, 632, 561, 276)

889² = 790321, 7² + 9² + 0² + 3² + 2² + 1² = 12²,
889² = 790321, 7 + 9 + 0 + 32 + 1 = 7²,
889² = 790321, 7 + 90 + 3 + 21 = 11²,
889² = 790321, 79 + 0 + 321 = 20².

Page of Squares : First Upload December 26, 2005 ; Last Revised August 29, 2011
by Yoshio Mimura, Kobe, Japan

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